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Growth in permutation groups and linear algebraic groups H. A. Helfgott Introduction Diameter bounds Growth in permutation groups and linear New work on algebraic groups permutation groups H. A. Helfgott September 2018

Growth in Cayley graphs permutation groups and linear algebraic groups H. A. Helfgott Definition Introduction G = � S � is a group. The (undirected) Cayley graph Diameter bounds New work on Γ( G , S ) has permutation groups vertex set G and edge set {{ g , ga } : g ∈ G , a ∈ S } .

Growth in Cayley graphs permutation groups and linear algebraic groups H. A. Helfgott Definition Introduction G = � S � is a group. The (undirected) Cayley graph Diameter bounds New work on Γ( G , S ) has permutation groups vertex set G and edge set {{ g , ga } : g ∈ G , a ∈ S } . Definition The diameter of Γ( G , S ) is k g = s 1 · · · s k , s i ∈ S ∪ S − 1 . diam Γ( G , S ) = max g ∈ G min (Same as graph theoretic diameter.)

Growth in How large can the diameter be? permutation groups and linear algebraic groups H. A. Helfgott The diameter can be very small: Introduction diam Γ( G , G ) = 1 Diameter bounds New work on permutation groups

Growth in How large can the diameter be? permutation groups and linear algebraic groups H. A. Helfgott The diameter can be very small: Introduction diam Γ( G , G ) = 1 Diameter bounds New work on permutation groups The diameter also can be very big: G = � x � ∼ = Z n , diam Γ( G , { x } ) = ⌊ n / 2 ⌋ More generally, G with a large abelian quotient may have Cayley graphs with diameter proportional to | G | .

Growth in How large can the diameter be? permutation groups and linear algebraic groups H. A. Helfgott The diameter can be very small: Introduction diam Γ( G , G ) = 1 Diameter bounds New work on permutation groups The diameter also can be very big: G = � x � ∼ = Z n , diam Γ( G , { x } ) = ⌊ n / 2 ⌋ More generally, G with a large abelian quotient may have Cayley graphs with diameter proportional to | G | . For generic G , however, diameters seem to be much smaller than | G | .

Growth in How large can the diameter be? permutation groups and linear algebraic groups H. A. Helfgott The diameter can be very small: Introduction diam Γ( G , G ) = 1 Diameter bounds New work on permutation groups The diameter also can be very big: G = � x � ∼ = Z n , diam Γ( G , { x } ) = ⌊ n / 2 ⌋ More generally, G with a large abelian quotient may have Cayley graphs with diameter proportional to | G | . For generic G , however, diameters seem to be much smaller than | G | . Example: for the group G of permutations of the Rubik cube and S the set of moves, | G | = 43252003274489856000, but diam ( G , S ) = 20 (Davidson, Dethridge, Kociemba and Rokicki, 2010)

Growth in The diameter of groups permutation groups and linear algebraic groups Definition H. A. Helfgott diam ( G ) := max diam Γ( G , S ) Introduction S Diameter bounds New work on permutation groups

Growth in The diameter of groups permutation groups and linear algebraic groups Definition H. A. Helfgott diam ( G ) := max diam Γ( G , S ) Introduction S Diameter bounds New work on Conjecture (Babai, in [Babai,Seress 1992]) permutation groups There exists a positive constant c : such that G finite, simple, nonabelian ⇒ diam ( G ) = O (log c | G | ) .

Growth in The diameter of groups permutation groups and linear algebraic groups Definition H. A. Helfgott diam ( G ) := max diam Γ( G , S ) Introduction S Diameter bounds New work on Conjecture (Babai, in [Babai,Seress 1992]) permutation groups There exists a positive constant c : such that G finite, simple, nonabelian ⇒ diam ( G ) = O (log c | G | ) . Conjecture true for PSL ( 2 , p ) , PSL ( 3 , p ) (Helfgott 2008, 2010) PSL ( 2 , q ) (Dinai; Varjú); work towards PSL n , PSp 2 n (Helfgott-Gill 2011) groups of Lie type of bounded rank (Pyber, E. Szabó 2011) and (Breuillard, Green, Tao 2011) But what about permutation groups? Hardest: what about the alternating group A n ?

Growth in Alternating groups, Classification Theorem permutation groups and linear algebraic groups H. A. Helfgott Introduction Reminder: a permutation group is a group of Diameter bounds permutations of n objects. New work on permutation S n = group of all permutations (S = “symmetric”) groups A n = unique subgroup of S n of index 2 (A = “alternating”)

Growth in Alternating groups, Classification Theorem permutation groups and linear algebraic groups H. A. Helfgott Introduction Reminder: a permutation group is a group of Diameter bounds permutations of n objects. New work on permutation S n = group of all permutations (S = “symmetric”) groups A n = unique subgroup of S n of index 2 (A = “alternating”) An asymptotic person’s view of the Classification Theorem:

Growth in Alternating groups, Classification Theorem permutation groups and linear algebraic groups H. A. Helfgott Introduction Reminder: a permutation group is a group of Diameter bounds permutations of n objects. New work on permutation S n = group of all permutations (S = “symmetric”) groups A n = unique subgroup of S n of index 2 (A = “alternating”) An asymptotic person’s view of the Classification Theorem: The finite simple groups are (a) finite groups of Lie type, (b) A n , (c) a finite number of unpleasant things (“sporadic”).

Growth in Alternating groups, Classification Theorem permutation groups and linear algebraic groups H. A. Helfgott Introduction Reminder: a permutation group is a group of Diameter bounds permutations of n objects. New work on permutation S n = group of all permutations (S = “symmetric”) groups A n = unique subgroup of S n of index 2 (A = “alternating”) An asymptotic person’s view of the Classification Theorem: The finite simple groups are (a) finite groups of Lie type, (b) A n , (c) a finite number of unpleasant things (“sporadic”).

Growth in Alternating groups, Classification Theorem permutation groups and linear algebraic groups H. A. Helfgott Introduction Reminder: a permutation group is a group of Diameter bounds permutations of n objects. New work on permutation S n = group of all permutations (S = “symmetric”) groups A n = unique subgroup of S n of index 2 (A = “alternating”) An asymptotic person’s view of the Classification Theorem: The finite simple groups are (a) finite groups of Lie type, (b) A n , (c) a finite number of unpleasant things (“sporadic”). Finite numbers of things do not matter asymptotically. We can thus focus on (a) and (b).

Growth in Diameter of the alternating group: results permutation groups and linear algebraic groups H. A. Helfgott Theorem (Helfgott, Seress 2011) Introduction Diameter bounds diam ( A n ) ≤ exp( O (log 4 n log log n )) . New work on permutation groups

Growth in Diameter of the alternating group: results permutation groups and linear algebraic groups H. A. Helfgott Theorem (Helfgott, Seress 2011) Introduction Diameter bounds diam ( A n ) ≤ exp( O (log 4 n log log n )) . New work on permutation groups Corollary G ≤ S n transitive ⇒ diam ( G ) ≤ exp( O (log 4 n log log n )) .

Growth in Diameter of the alternating group: results permutation groups and linear algebraic groups H. A. Helfgott Theorem (Helfgott, Seress 2011) Introduction Diameter bounds diam ( A n ) ≤ exp( O (log 4 n log log n )) . New work on permutation groups Corollary G ≤ S n transitive ⇒ diam ( G ) ≤ exp( O (log 4 n log log n )) . The corollary follows from the main theorem and (Babai-Seress 1992), which uses the Classification. (As pointed out by Pyber, there is an error in (Babai-Seress 1992), but it has been fixed.)

Growth in Diameter of the alternating group: results permutation groups and linear algebraic groups H. A. Helfgott Theorem (Helfgott, Seress 2011) Introduction Diameter bounds diam ( A n ) ≤ exp( O (log 4 n log log n )) . New work on permutation groups Corollary G ≤ S n transitive ⇒ diam ( G ) ≤ exp( O (log 4 n log log n )) . The corollary follows from the main theorem and (Babai-Seress 1992), which uses the Classification. (As pointed out by Pyber, there is an error in (Babai-Seress 1992), but it has been fixed.) The Helfgott-Seress theorem also uses the Classification.

Growth in Product theorems permutation groups and linear algebraic groups Since (Helfgott 2008), diameter results for groups of Lie H. A. Helfgott type have been proven by product theorems: Introduction Diameter bounds Theorem New work on permutation There exists a polynomial c ( x ) such that if G is simple, groups Lie-type of rank r, G = � A � then A 3 = G or | A 3 | ≥ | A | 1 + 1 / c ( r ) . In particular, for bounded r, we have | A 3 | ≥ | A | 1 + ε for some constant ε .